Algebraic Combinatorics and its Applications

组合数学是一个专门研究离散结构的数学领域，如图表，图形，各种组合，多面体等。一个典型的组合问题是“如何计算这些结构？”组合问题出现在纯数学、应用数学、物理学和其他科学的各个领域。 这个研究项目中的几个问题与基础物理学的最新进展有关。 这项研究将有助于更好地理解自然界的基本过程：基本粒子之间的相互作用。 组合数学的一个独特的特点是组合对象非常直观，易于理解。 许多组合问题可以解释给一个高中生。 PI和RA将在各种受众中传播研究成果。 PI将鼓励研究生、本科生和高中生研究组合问题。 他将支持麻省理工学院PRIMES“高中生数学，工程和科学研究计划”。 本研究计画将增进大众对组合数学与数学的认识，并致力于研究代数组合数学中的几个问题及其应用。 许多问题都涉及到正格拉斯曼，一个美丽的几何对象与丰富的组合结构。 在正格拉斯曼研究中发展起来的组合结构和技术也在其他领域出现：逆边界问题，拟阵理论，凸几何，环面几何，统计力学，孤立子理论，Fomin-Zelevinsky的簇代数，对称函数，仿射舒伯特微积分，Lusztig的标准基，矩阵完成问题，Schur正性问题，以及研究基本粒子的散射振幅-物理学中最基本的领域之一。 有几个问题与幂理想有关，幂理想是交换代数中与图、超平面排列、停车函数、拟阵和其他组合结构相联系的对象。 该项目涉及的问题广义置换面，这是某些凸多面体，其体积和整数格点的数量表示许多经典的组合序列：欧拉数，加泰罗尼亚数，树木和森林的数量，以及它们的推广。 他们与低维拓扑和热带定向拟阵的联系，并关注一个有趣的新的类似Tutte多项式。 该项目还包含了几类凸多面体的问题：根多面体，凹多面体和流多面体。

Combinatorics is the area of mathematics dedicated to the study of discrete structures, such as diagrams, graphs, various combinations, polytopes, etc. A typical combinatorial question is ``How to count these structures?'' Combinatorial problems appear in various areas of pure and applied mathematics, physics, and other sciences. Several problems in this research project are related to the latest advances in the fundamental physics. This study will lead to a better understanding of the fundamental processes of the nature: interactions between elementary particles. A unique feature of combinatorics is that combinatorial objects are quite visual and easy to understand. Many combinatorial problems can be explained to a high-school student. The PI and RAs will disseminate research results among various audiences. The PI will encourage graduate, undergraduate, and high-school students to work on the combinatorial problems. He will support MIT PRIMES "Program for Research in Mathematics, Engineering and Science for High School Students". This project will promote public knowledge of combinatorics and mathematics in general.This research project is devoted to the study of several problems in algebraic combinatorics and their applications. Many of the problems are related to the positive Grassmannian, a beautiful geometrical object with a rich combinatorial structure. The combinatorial structures and techniques that were developed in the study of the positive Grassmannian also surface in other areas: inverse boundary problems, matroid theory, convex geometry, toric geometry, statistical mechanics, theory of solitons, Fomin-Zelevinsky's cluster algebras, symmetric functions, affine Schubert calculus, Lusztig's canonical bases, matrix completion problems, Schur positivity problems, as well as the study of scattering amplitudes of elementary particles - one of the most fundamental field in physics. Several problems are related to power ideals, which are objects from commutative algebra linked with graphs, hyperplane arrangements, parking functions, matroids, and other combinatorial structures. The project involves problems on generalized permutohedra, which are certain convex polytopes, whose volumes and numbers of integer lattice points express many classical combinatorial sequences: the Eulerian numbers, the Catalan numbers, the numbers of trees and forests, and their generalizations. They have links with low-dimensional topology and tropical oriented matroids, and concern a interesting new analogue of the Tutte polynomial. The project also contains problems on several classes of convex polytopes: root polytopes, alcoved polytopes, and flow polytopes.